You are on vacation in San Francisco and decide to take a cable car to see the city. A 5500 kg cable car goes 360 m up a hill inclined 19 ∘ above the horizontal.
What can your learn about the work done by the cable to move car up the hill? Answer quantitatively. Ignore friction.
W = Fd
F = mg sin19
To determine the work done by the cable car to move up the hill, we will use the formula for work:
Work = force * distance * cos(angle)
In this case, the force can be determined using the weight of the cable car:
Force = mass * gravity
where mass = 5500 kg and gravity = 9.8 m/s^2 (acceleration due to gravity).
Next, we need to find the angle in radians, which is given as 19 ∘ above the horizontal. To convert this angle to radians, we use the following formula:
angle (radians) = angle (degrees) * π / 180
Substituting the given values:
angle (radians) = 19 ∘ * π / 180
Now, we can calculate the work done:
Work = (5500 kg * 9.8 m/s^2) * 360 m * cos(19 ∘ * π / 180)
Let's calculate this:
Work = (5500 * 9.8) * 360 * cos(19 * π / 180)
Using a calculator, we find:
Work ≈ 901576.74 J (Joules)
Therefore, the work done by the cable to move the car up the hill is approximately 901576.74 Joules.
To find out the work done by the cable car to move up the hill, we need to calculate the amount of energy transferred to the car. Work is defined as the force applied multiplied by the distance over which the force is applied. In this case, the force applied by the cable car is the weight of the car, and the distance is the vertical height the car moves.
First, let's find the magnitude of the force the cable car exerts on the car. Since the car is on an incline, we need to consider the component of the weight parallel to the hill.
The weight of the car can be calculated using the formula:
weight = mass * gravitational acceleration
Given:
mass (m) = 5500 kg
gravitational acceleration (g) = 9.8 m/s^2
weight = 5500 kg * 9.8 m/s^2 = 53900 N
Now, we need to find the component of the weight parallel to the hill. This can be calculated using trigonometry, specifically sine function.
force_parallel = weight * sin(angle)
Given:
angle = 19°
force_parallel = 53900 N * sin(19°) ≈ 17812.38 N
The work done by the cable car can be calculated using the formula:
work = force_parallel * distance
Given:
distance = 360 m
work = 17812.38 N * 360 m = 6413258.8 Joules
Therefore, the work done by the cable car to move the car up the hill is approximately 6,413,258.8 Joules.